One parameter subgroup

Question

I am new to Lie group and I am reading the "Lie Groups, Lie Algebras, and Representations" by Brian Hall. So what's the intuitive idea about one parameter subgroup? I understand all the definition but lack a general big picture about it. And why it is sometimes called the infinitesimal description of Lie group?

Answer 1

A 1-parameter subgroup on $G$ is simply a Lie group homomorphism from $\mathbb{R}$ to $G$. I just picture it as a curve through the identity.

There is a bijection between the space of 1-parameter subgroups and the tangent space at the identity, given by $\theta\mapsto\theta_*\left(\left.\frac{\partial}{\partial t}\right|_0\right)$. The tangent space at the identity can be thought of as consisting of "infinitesimal generators" of the group, because $$G_\mathrm{e}=\left<\gamma_{v_1}(1)\cdots\gamma_{v_n}(1)~\vert~(\gamma_{v_i})_*\left(\left.\frac{\partial}{\partial t}\right|_0\right)=v_i\right>,$$ where $G_\mathrm{e}$ is the path component of $G$ containing the identity.